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2017 AMC 10B Problem 9

Problem 9 of 25EasierProbability & StatisticsProblem-Solving Techniques

A radio program has a quiz consisting of 33 multiple-choice questions, each with 33 choices. A contestant wins if he or she gets 22 or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?

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Solution

The probability that a contestant gets all 33 correct is (13)3=127.\left(\dfrac13\right)^3=\dfrac1{27}. The probability of getting exactly 22 correct is (32)(13)2(23)=627.\binom32\left(\dfrac13\right)^2\left(\dfrac23\right)=\dfrac6{27}. The combined probability is 627+127=727.\dfrac6{27}+\dfrac1{27}=\dfrac7{27}. Thus, the correct answer is D .
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